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Section 3.1: Number Representation
Practice HW (not to hand in)
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Representation of Numbers
• Ordinary numbers we see every day are
represented as base 10, that is, as the sum of
the powers of 10. We illustrate how this is so in
the following example.
• So far, we have studied historical cryptographic
methods that are typically hand-based. Modern
cryptographic methods are computer based
methods. To have an appreciation for these
methods, we need to understand how
computers communicate. In this section, we
study the types of numbers that computers
typically work with.
Example 1: Write the number 12341 as a sum of
the powers of 10.
Solution:
Binary Numbers are base 2 numbers are made
up only of 0’s and 1’s. Computers use these
numbers to represent data internally. Examples of
binary numbers are 0 (which represents the
number 0) 100 (which represents the number 4),
1001 (which represents the number 9), and
1011000 (which represents the number 88). We
now give a formal definition of a binary number.
Definition
A binary number bk bk 1 b2b1b0, where
bi  0 or 1, represents the base 10 decimal
number given by
bk 2  bk 1 2
k
k 1
   b2 2  b1 2  b0 2
2
1
0
We illustrate this definition in the following
examples.
Example 2: Find the base 10 decimal
representation of the binary numbers
a. 100
Solution:
b. 1011000
Solution:
c. 1110001011
Solution:
Note
• To convert a decimal (base 10) number to
binary, we compute the powers of 2 (starting
with 20 ) that are less than the given number.
Then write the number as a sum of these
powers of 2 from largest to smallest, writing a
coefficient of 1 in front the power of 2 that
occurs in the sum and a 0 in front of the power
of 2 that does not occur. Reading off the
coefficients from left to right gives the binary
representation. We illustrate this technique in
the following examples.
Example 3: Convert 77 to binary.
Solution:
Example 4: Convert 320 to binary.
Solution:
Approximating the Size of a Binary Number
• Many times the strength of a cryptographic
method is expressed in terms of the size of a
particular parameter. Many times the size of this
parameter is expressed in terms on the number
of binary digits the number has. The following
formula gives an estimate of the number of
binary digits is required to expressed a given
base 10 decimal number.
Number of Binary Digits Estimate
ln( n)
b
ln( 2)
where
n  base 10 decimal number
b  approximat e number of binary digits in the given number
Example 5: Approximate how many binary digits
are used to represent the number 430121.
Solution:
• The following inequality gives a bound of the
size of a base 10 number given the number of
binary digits that represent it.
Base 10 Number Size Binary Bound Estimate
b 1
2
 n  2 , where
b
n  base 10 decimal number
b  approximat e number of binary digits in the given number
Example 6: A base 10 number is represented by
a 32 bit number. Find a bound that will estimate
its size.
Solution:
ASCII Codes for Characters
• So far, we have used the MOD 26 alphabet
assignment table to assign a numerical
representation to each letter. Computers
normally use the ASCII (American Standard
Code for Information Interchange) table for
obtaining numerical representation of
characters.
Table 1: ASCII table for characters
Example 7: Find the numerical Ascii table
representation for the characters z, Z, “, and a
space.
Solution:
Example 8: Decode the following text
represented by the 8-bit blocks representing Ascii
code numbers:
01001101
01001111
01000100
00110010
Solution: